 
Summary: VARIETIES OF COMPLETELY DECOMPOSABLE FORMS AND
THEIR SECANTS
HIROTACHI ABO
Abstract. This paper is devoted to the study of higher secant varieties of va
rieties of completely decomposable forms. The main goal is to develop methods
to inductively verify the nondefectivity of such secant varieties. As an appli
cation of these methods, we will establish the existence of large families of
nondefective secant varieties of "small" varieties of completely decomposable
forms.
1. Introduction
In 1770, E. Waring suggested the problem of finding a positive integer s such
that every positive integer can be written as the sum of at most s dth
power of
positive integers. In 1909, this problem was solved affirmatively by Hilbert. There
is a polynomial version of this problem, which asks, "What is the smallest integer
s such that a general dform in (n + 1) variables is expressible as the sum of s
dth
powers of linear forms?" This problem is often called Waring's problem for
polynomials. Waring's problem for polynomials has remained unsolved for many
years, but was completed in a series of papers [3, 2, 1] by Alexander and Hirschowitz
